Compact group topologies for 𝑅

Hawley, Douglas

doi:10.1090/S0002-9939-1971-0281834-5

Compact group topologies for $R$
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by Douglas Hawley

Proc. Amer. Math. Soc. 30 (1971), 566-572

DOI: https://doi.org/10.1090/S0002-9939-1971-0281834-5

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Abstract:

There is a compact solenoidal group S that is group isomorphic to the additive real numbers R. The existence of S leads to a study of compact real groups; that is, compact groups which are group isomorphic to R. The compact real groups are characterized as products of S. Various conditions on a topological group G are given which are equivalent to G being compact real. It is shown that any compact real group is solenoidal, connected and separable. Many sets and functions on R which are measurable with respect to the usual topology are shown to be nonmeasurable with respect to any compact group topology.

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